Summation of rational series twisted by strongly B-multiplicative coefficients

Abstract

We evaluate in closed form series of the type $\sum u(n)R(n)$, where $(u(n){)}_n$ is a strongly $B$-multiplicative sequence and $R(n)$ a (well-chosen) rational function. A typical example is: $$\sum _{n\ge 1}(-1{)}^{s_2(n)}\frac{4n+1}{2n(2n+1)(2n+2)}=-\frac{1}{4}$$ where $s_2(n)$ is the sum of the binary digits of the integer $n$. Furthermore closed formulas for series involving automatic sequences that are not strongly $B$-multiplicative, such as the regular paperfolding and Golay-Shapiro-Rudin sequences, are obtained; for example, for integer $d\ge 0$: $$\sum _{n\ge 0}\frac{v(n)}{(n+1{)}^{2d+1}}=\frac{{\pi }^{2d+1}|E_{2d}|}{(2^{2d+2}-2)(2d)!}$$ where $(v(n){)}_n$ is the $\pm 1$ regular paperfolding sequence and $E_{2d}$ is an Euler number.

Typo in a crossreference corrected in Example 9, page 6. Remark added top of Page 9 about the relation between paperfolding and the Jacobi-Kronecker symbol

Cited by in corpus (1)

• Errata and Addenda to Mathematical Constants